# Solving Inequalities Using Four Basic Operations

## Solving Inequality Using Four Basic Operations

Here, we’re going to look at a slightly more complex inequality. **Recall the rules for solving any inequality are very similar to the rules for solving an equation.** You can manipulate the inequality by performing operations, so long as you perform the same operations to both sides of the inequality.

**There is, however, one additional provision for inequalities.** If you multiply or divide both sides of the inequality by a negative number, you have to reverse the sign on the inequality. For instance, if I were to multiply both sides of this inequality by -1, I would have to change this less than sign to a greater than sign.

**Let’s keep that in mind as we solve this.** Now, the first thing we want to do here is eliminate the denominator on this fraction, so that we can just have whole numbers and whole number coefficients. **What we’re going to do is simplify the denominator and then multiply everything through by the denominator.**

What we have here is 8 minus 12, which is equivalent to -4. We need to multiply each term in this inequality by -4. We can multiply- if we multiply this fraction by -4, we just eliminate the denominator. What we’re left with is 5x plus 2. Multiply 3x by -4 and we get -12x Minus 12x. We multiply 10x by -4 and we get -40x.

We multiply -1 by -4 and we get a positive 4. Note that we multiplied by a negative number, so we have to reverse the sign on the inequality. **This is now going to become a greater than sign.** Now, all we have to do is isolate the x on one side of the equation and solve for it. We need to add 40x to both sides.

That’ll eliminate x from the right side. We’ll subtract 2 from both sides. That will get rid of the twos and get rid of the 40x’s. Now we just need to combine our x’s. We have 5 minus 12 plus 40, which is going to be 33x, which is greater than 4 minus 2 is 2. We know that x is greater than 2/33. That is going to be our solution to this inequality.

Let’s make sure by plugging back in a value for x that is greater than 2/33 just to make sure that this inequality is correct. 1 is greater than 2/33, so we’ll plug in 1. What we’ve got here, if we plug in 1, is we’ve got 5 plus 2 over 8 minus 12 plus 3 is less than 10 minus 1. We’ll simplify this. 7/-4 plus 3 is less than 9. 7/-4 is -1 3/4 plus 3 is going to be positive 1 1/4. 1 1/4 is less than 9. This inequality holds correct.